# Definition:General Logarithm/Common

## Contents

## Definition

Logarithms base $10$ are often referred to as common logarithms.

### Notation for Negative Logarithm

Let $n \in \R$ be a real number such that $0 < n < 1$.

Let $n$ be presented (possibly approximated) in scientific notation as:

- $a \times 10^{-d}$

where $d \in \Z_{>0}$ is a (strictly) positive integer.

Let $\log_{10} n$ denote the common logarithm of $n$.

Then it is the standard convention to express $\log_{10} n$ in the form:

- $\log_{10} n = \overline d \cdotp m$

where $m := \log_{10} a$ is the mantissa of $\log_{10} n$.

## Mantissa

Let $\log_{10} n$ be expressed in the form:

- $\log_{10} n = \begin {cases} c \cdotp m & : d \ge 0 \\ \overline c \cdotp m & : d < 0 \end {cases}$

where:

- $c = \size d$ is the absolute value of $d$
- $m := \log_{10} a$

$\log_{10} a$ is the **mantissa** of $\log_{10} n$.

## Characteristic

$c$ is the **characteristic** of $\log_{10} n$.

## Examples

### Common Logarithm: $\log_{10} 1000$

The common logarithm of $1000$ is:

- $\log_{10} 1000 = 3$

### Common Logarithm: $\log_{10} 0 \cdotp 01$

The common logarithm of $0 \cdotp 01$ is:

- $\log_{10} 0 \cdotp 01 = -2$

### Common Logarithm: $\log_{10} 2$

The common logarithm of $2$ is:

- $\log_{10} 2 \approx 0.30102 \, 99956 \, 63981 \, 19521 \, 37389 \ldots$

### Common Logarithm: $\log_{10} 3$

The common logarithm of $3$ is:

- $\log_{10} 3 = 0.47712 \, 12547 \, 19662 \, 43729 \, 50279 \ldots$

### Common Logarithm: $\log_{10} e$

The common logarithm of Euler's number $e$ is:

- $\log_{10} e = 0 \cdotp 43429 \, 44819 \, 03251 \, 82765 \, 11289 \, 18916 \, 60508 \, 22943 \, 97005 \, 803 \ldots$

### Common Logarithm: $\log_{10} \pi$

The common logarithm of $\pi$ is:

- $\log_{10} \pi = 0.49714 \, 98726 \, 94133 \, 85435 \, 12683 \ldots$

## Also known as

**Common logarithms** are sometimes referred to as **Briggsian logarithms** or **Briggs's logarithms**, for Henry Briggs.

In elementary textbooks and on most pocket calculators, $\log$ is assumed to mean $\log_{10}$.

This ambiguous notation is not recommended, particularly since $\log$ often means base $e$ in more advanced textbooks.

## Also see

- Results about
**logarithms**can be found here.

## Historical Note

**Common logarithms** were developed by Henry Briggs, as a direct offshoot of the work of John Napier.

After seeing the tables that Napier published, Briggs consulted Napier, and suggested defining them differently, using base $10$.

In $1617$, Briggs published a set of tables of logarithms of the first $1000$ positive integers.

In $1624$, he published tables of logarithms which included $30 \, 000$ logarithms going up to $14$ decimal places.

Before the advent of cheap means of electronic calculation, **common logarithms** were widely used as a technique for performing multiplication.

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 7$: Common Logarithms and Antilogarithms - 1972: Murray R. Spiegel and R.W. Boxer:
*Theory and Problems of Statistics*(SI ed.) ... (previous) ... (next): Chapter $1$: Logarithms - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $10$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $10$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Briggsian logarithm** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Briggsian logarithm** - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $5$: Eternal Triangles: Logarithms